5d fixed points from brane webs and O7-planes

We explore the properties of five-dimensional supersymmetric gauge theories living on 5-brane webs in orientifold 7-plane backgrounds. These include $USp(2N)$ and $SO(N)$ gauge theories with fundamental matter, as well as $SU(N)$ gauge theories with symmetric and antisymmetric matter. We find a number of new 5d fixed point theories that feature enhanced global symmetries. We also exhibit a number of new 5d dualities.


Introduction
Although 5d gauge theories are perturbatively non-renormalizable, in many N = 1 supersymmetric cases they are UV complete [1][2][3]. Namely, there exist interacting 5d N = 1 superconformal theories with relevant deformations corresponding to an inverse Yang-Mills coupling of a 5d N = 1 supersymmetric gauge theory. The superconformal theory is in some sense the infinite coupling limit of the gauge theory. The existence of such fixed point theories is also suggested by the existence of a (unique) 5d superconformal algebra F (4) (which has a bosonic subgroup SO(5, 2) × SU (2) R ). However these UV fixed point theories do not admit a Lagrangian description.
A useful way to describe such theories is using 5-brane webs in Type IIB string theory [4,5]. This realizes a 5d SCFT as an intersection of 5-branes at a point. The moduli and mass parameters of the SCFT are described as motions of the internal and external 5-branes, respectively. In particular, the 5d SCFT may posses a deformation leading to a low-energy gauge theory described by a simple 5-brane configuration containing stacks of D5-branes. This allows us to study various aspects of 5d superconformal theories and the associated gauge theories.
First, it provides a new way to classify 5d gauge theories that have UV fixed points, since the gauge groups and matter content are constrained by the brane construction. In some cases this extends the perturbative classification of [3].
5-brane webs also allow us to uncover dual gauge theories in five dimensions. Since in 5d masses are real, the IR theories obtained by positive and negative mass deformations may be different. In particular these may be different gauge theories related by a continuation of the Yang-Mills coupling past infinity. This can be seen in the 5-brane web construction by reversing the deformation leading to the original gauge theory, and using the SL(2, Z) symmetry of Type IIB string theory to transform the web into a configuration with D5brane stacks.
Finally, 5-brane webs can also motivate and assist in demonstrating non-perturbatively enhanced global symmetries. 5d gauge theories possess U (1) global symmetries associated to the instanton number currents, j I = * Tr(F ∧ F ), in each non-abelian gauge group factor. In some cases the instanton operators associated to these currents provide additional conserved currents, and lead to a larger global symmetry than is apparent in the gauge theory Lagrangian. Such is the case, for example, for the SU (2) theory with N F ≤ 7, where the classical SO(2N F ) × U (1) I symmetry is enhanced by instantons to E N F +1 [1]. Since conserved currents belong to BPS multiplets of operators, enhancement of global symmetries is exhibited by the 5d superconformal index, which can in turn be computed for the IR gauge theory using localization [6][7][8][9]. The crucial ingredient is of course the instanton contribution, which can be obtained from known expressions for instanton partition functions in four dimensions. However in some cases corrections are required [7,[9][10][11]. In the 5-brane web description these corrections are identified with extraneous states, corresponding to strings external to the web whose contributions have to be removed by hand from the partition function.
Most 5-brane web constructions so far have been for theories with SU (N ) gauge groups and matter fields in fundamental or bi-fundamental representations 1 . Our main purpose in this paper is to extend the study of 5d N = 1 gauge theories and fixed point theories via 5-brane webs to theories with U Sp(2N ) and SO(N ) gauge groups, as well as to SU (N ) theories with rank-two antisymmetric and symmetric matter. We will do this by including orientifold 7-planes. We will exhibit 5d fixed point theories, some of them new, with relevant deformations given by gauge theories of the above form. We will also find new 5d dualities, and examples of enhanced global symmetry involving the above types of theories.
The outline for the rest of the paper is as follows. Section 2 is devoted to U Sp(2N ) theories, constructed using 5-brane webs in an O7 − plane background. In section 3 we  construct SO(M ) theories using an O7 + plane. In sections 4 and 5 we construct SU (N ) theories with rank 2 antisymmetric and symmetric matter, by adding a fractional NS5brane to the O7 − and O7 + plane, respectively. We conclude in section 6. We also include an appendix containing a discussion of corrections to instanton partition functions in the relevant cases of U Sp(2N ) and SO(M ).

O7 − and U Sp gauge theories
A classical 5-brane web configuration for a pure U Sp(2N ) theory (without an antisymmetric matter field) is shown in Fig. 1a. This includes an orientifold 7-plane of type O7 − , which is parallel to the various 7-branes on which the external 5-branes end. 2 This is essentially a 5d generalization of the 4d constructions in [13][14][15].
In the figure we show the covering space, including two copies of the reduced web related by a reflection across the origin. We have also included two copies of the cut associated with the monodromy of the O7 − plane, The physical space is the upper half plane, with the left and right halves of the cut identified. The discontinuity of (p, q)5-brane charges across the cut corresponds to a clockwise action of the monodromy. The bare 5d Yang-Mills coupling g −2 0 is given by the separation along the cut.
We are still left with a choice of an integer k, corresponding to the D5-brane charge of one external 5-brane. Naively the different choices are all related by T ∈ SL(2, Z), which shifts k → k + 1, and therefore all describe the same U Sp(2N ) gauge theory. On the other hand, we know that the U Sp(2N ) theory admits a discrete theta parameter associated with π 4 (U Sp(2N )) = Z 2 . Apparently, the O7 − plane is not invariant under T , but only under T 2 . The theta parameter is then related to the parity of k. We will say more about this below.
At the quantum level, the O7 − plane is resolved into a pair of mutually non-local (p, q) 7-branes, whose combined monodromy is given by that of the O7 − plane [16]. 3 There is some ambiguity in the choice of the (p, q) charges of the two 7-branes. In particular, the element T ∈ SL(2, Z) transforms (p, q) → (p + q, q), but clearly leaves the total monodromy M O7 − = −T −4 invariant. One common choice for the (p, q) charges of the 7-branes is {(2, 1), (0, −1)}. Another, related by T , is {(1, 1), (1, −1)}. We will now argue that these two resolutions correspond to physically distinct O7 − planes, and more generally that there are two variants of the O7 − plane related by T .
Taking the 7-brane charges to be {(1, 1), (1, −1)}, we get the brane web shown in Fig. 1b (in the reduced space). Note the change in the 5-brane charges across the two cuts. We can get a simpler presentation by moving the 7-branes outside. Accounting for brane creation and for the effect of moving the cuts, we end up with the 5-brane web shown in Fig. 1c. In this presentation the fact that the gauge group is U Sp(2N ), rather than say SU (2N ), is not immediately obvious. It follows from the constraint imposed on the N -junctions by the s-rule. We can now understand the connection between k and the theta parameter. Consider the N = 1 case, namely the U Sp(2), or SU (2), theory. We identify the k = 1 web with the θ = π theory, and the k = 2 web with the θ = 0 theory. Furthermore, shifting k → k + 2 leaves θ invariant.
If we instead resolve the orientifold plane into 7-branes with charges {(2, 1), (0, −1)} the resulting web would be different. Acting with T brings it to the form of the web in Fig. 1c, but with k → k + 1. This describes the U Sp(2N ) theory with the other value of θ. We conclude from this that there are two physically distinct variants of the O7 − plane, one of which is resolved to a 7-brane pair with charges {(1, 1), (1, −1)} up to an action of T 2n , and the other to a pair with charges {(2, 1), (0, −1)} up to an action of T 2n . It would be interesting to see this directly from the point of view of the orientifold. This is presumably also related to the transition between the two values of θ in the Type I' brane construction given in [17].

Flavors
Matter in the fundamental representation (flavor) can be added by attaching external D5branes. Requiring that external 5-branes do not intersect (which would lead to additional massless degrees of freedom) leads to the condition that N F ≤ 2N + 4 (Fig. 2a), in agreement with the classification of [3]. We claim however that also with N F = 2N + 5 one remains in the realm of consistent 5d theories. In particular, the N = 1 case is the rank one E 8 theory. 4 The 5-brane web for N F = 2N + 5 is shown in Fig. 2b. The dangerous 3 The monodromy associated with a (p, q)7-brane is given by intersection is avoided as a consequence of the s-rule. This is similar to the situation for SU (N ) with N F = 2N + 1 [18], which is also one more flavor than allowed by [3]. As before, the O7 − -plane is resolved quantum mechanically into a pair of 7-branes, and one can obtain an alternative 5-brane web realization of the theories by Hanany-Witten transitions. In particular for N = 1 we get the familiar 5-brane webs for SU (2) with N F = 1, 2, 3 and 4 flavors, as well as those describing N F = 5, 6 and 7 flavors (the E 6 , E 7 and E 8 theories) [19]. The 5-brane webs shown in Fig. 2 suggest that the global symmetry is enhanced at the fixed points in these cases. For N F = 2N + 4 the classical global symmetry of the IR gauge theory is SO(4N + 8) F × U (1) I . The pair of parallel external legs suggests that the U (1) I factor is enhanced to SU (2), like in the case of SU (N ) 0 with N F = 2N [7]. For N F = 2N + 5 on the other hand, the parallel external legs, D5-branes in this case, suggest that the classical global symmetry SO(4N + 10) F × U (1) I is enhanced to SO(4N + 12).
This can be seen explicitly in the 5d superconformal index, the key ingredient of which is the contribution of instanton states corresponding to the 5d lift of the multi-instanton partition function [6]. In the above two cases the instanton computation exhibits the same kind of pathologies associated with extraneous "decoupled" states that were encountered in other cases in [7,18], such as for SU (N ) with N F = 2N and N F = 2N + 1. The relevant states in the present cases are shown in red in Fig. 2. For N F = 2N + 4 it is a D-string between the external parallel NS5-branes, and for N F = 2N + 5 there is a fundamental string between the flavor D5-branes (or their images) and the extra external D5-brane, and a 3-pronged string attached to the color D5-branes (or their images). Note that all of these carry two units of instanton charge. 5 This follows from the fact that the D-string worldvolume gauge symmetry is O(2), which is the ADHM dual group for two instantons of U Sp(2N ). The minimally instanton-charged object corresponds to a fractional D-string that intersects the O7-plane, and cannot move away from it. The string states in the N F = 2N + 5 web are also charged under other symmetries. The fundamental string state is in the vector representation of SO(4N + 10) F , and the 3-pronged string is in the fundamental representation of U Sp(2N ).
The contribution of these states to the 2-instanton term then has the form where x and y are the fugacities associated with the Cartans of SO(4) ⊂ SO (5,2). This must be subtracted from the 2-instanton partition function to obtain a consistent result (see Appendix for details). In the case of N F = 2N + 4 we expect the subtraction to plethystically exponentiate to a correction factor for the full multi-instanton partition function 6 We verify this up to the 4-instanton level in the Appendix. In the N F = 2N + 5 case, the presence of the gauge-charged state complicates the counting, and we can only perform the subtraction at the basic 2-instanton level.
Combining the corrected instanton partition functions with the perturbative contributions from the gauge and matter supermultiplets we find that the superconformal indices are given by: 5) and The x 2 terms correspond to the contributions of conserved current multiplets. We see that the 2-instanton states provide additional charged currents. For N F = 2N + 4 these lead to an enhancement of U (1) I to SU (2), and for N F = 2N + 5, of SO(4N + 10) F × U (1) I to SO(4N + 12).

Duality
Since we do not have a simple (perturbative) description for the S-dual of the O7-plane, we cannot identify the S-dual gauge theory directly in these cases. However for N = 1 we already know the answer in some cases. The U Sp(2) = SU (2) theory with N F ≤ 7 is a self-dual gauge theory description of the E N F +1 fixed point theory [20]. We can use this to determine the duals in higher rank cases. Let us consider the specific class of theories with gauge group U Sp(2N ) and N F = 2N + 2 fundamental hypermultiplets. This is an interesting class of examples, since it is related by dimensional reduction, as in [18], to a duality in four dimensions.   Figure 4: The orientifold 5-brane web for SU (2) + 6 and its S-dual, which also describes SU (2) + 6 (we have taken k = 0).
The simplest interesting case is U Sp(4) with N F = 6. The orientifold 5-brane web for this theory is shown in Fig. 3a. The S-dual web, Fig. 3b, corresponds to the same UV fixed point, but describes a different IR gauge theory. The dual theory appears to be a linear quiver with gauge group SU (2) × SU (2), but the matter content is not obvious. We can identify it more precisely by "ungauging" the first SU (2) factor, leading to the web in Fig. 4a. The S-dual of this web, Fig. 4b, corresponds to SU (2) with N F = 6. Since this theory is self-dual, the original web also describes SU (2) with N F = 6, albeit with only an SU (2) ⊂ SO(12) F exhibited manifestly. The dual of U Sp(4) with N F = 6, Fig. 3b, therefore corresponds to a gauging of SU (2) ⊂ SO(12) F in the SU (2), N F = 6 theory. The resulting gauge theory is the linear quiver theory SU (2) π × SU (2) + 4. The remaining matter global symmetry is SO(8) F × SU (2) BF , where SU (2) BF is associated to the bifundamental hypermultiplet. 7 We can therefore describe the 5d SCFT corresponding to the webs of Fig. 3 as gauging an SU (2) ⊂ E 7 in the rank one E 7 theory and flowing to the UV. The dual gauge theory descriptions of the resulting rank two SCFT correspond to different embeddings of SU (2) in E 7 , in one case leaving SO(12) F and in the other This is actually the 5d lift of the second 4d N = 2 duality of Argyres and Seiberg [22], which relates the strong coupling limit of the 4d U Sp(4) + 6 superconformal gauge theory to the weak gauging of an SU (2) ⊂ E 7 in the Minahan-Nemeschansky E 7 theory [23].
Generalizing to U Sp(2N ) with N F = 2N + 2 is straightforward (see Fig. 5). The S- Figure 5: The orientifold 5-brane web for U Sp(2N ) + 2N + 2 and its S-dual (k = 0). dual web corresponds to the linear quiver theory SU (2) π × SU (2) N −2 0 × SU (2) + 4. The shared UV fixed point is a 5d rank N SCFT with a (2N + 3)-dimensional space of mass parameters, and corresponds to gauging an SU (2) inside a particular 5d rank N − 1 SCFT. The latter has dual IR descriptions as the quiver theory 2 The reduction to 4d then gives a duality between the 4d N = 2 superconformal gauge theory with gauge group U Sp(2N ) and N F = 2N +2, and an SU (2) gauging of a particular isolated 4d N = 2 SCFT given by the dimensional reduction of the above 5d rank N − 1 SCFT. We can identify the 4d SCFT according to the classification of [24] in terms of a 3-punctured sphere with a specific set of punctures, by manipulating the 5-brane web to a standard triple 5-brane junction form, as shown in Fig. 6. This represents the theory as a limit of the T 2N theory, in which two of the maximal punctures (corresponding to the fully symmetrrized 2N -box Young tableau) are replaced by non-maximal punctures; an (N, N ) puncture (corresponding to a Young diagram with two columns of N boxes), and an (N − 1, N − 1, 1, 1) puncture. This 4d duality first appeared in [25].

O7 + and SO gauge theories
Replacing the O7 − plane with an O7 + plane, we obtain a 5-brane web for an SO(2N ) gauge theory. The monodromy matrix of the O7 + plane is given by M O7 + = −T 4 . The 5-brane web for the pure SO(2N ) theory is shown in Fig. 7a. This is the exact quantum configuration; unlike the O7 − -plane, the O7 + -plane is not resolved into 7-branes. As before, k → k + 1 under T . Since there is no additional parameter in the SO(2N ) theory, the web must describe the same theory for all k. 8 This implies that, unlike the O7 − plane, the O7 + plane must be invariant under T . A slight generalization of this web including a fractional D5-brane gives the pure SO(2N + 1) theory (Fig. 7b).

Flavors
The condition in [3] for an SO(M ) gauge theory to come from a 5d fixed point is N V ≤ M − 4. In the 5-brane webs this is again seen as the condition of no intersections. But, as in other cases, the 5-brane web construction appears to imply that one additional flavor is allowed.
As before, the 5-brane webs in these cases exhibit extraneous instanton-charged states (shown in red in Fig. 8) that are not part of the 5d gauge theories, and whose contribution to the instanton partition function must be removed by hand. Here they carry the minimal instanton charge, since there is no fractional BPS D-string. This can also be understood from the fact that the D-string worldvolume gauge symmetry is SU (2), which is the ADHM dual gauge group for one instanton of SO(M ).
For SO(M ) with N V = M − 4, the proposed correction factor is identical to the one in (2.4), except with q 2 replaced by q. This is verified in the Appendix to the 2-instanton level. In special cases one can also compare with other approaches. For SO(4) we compared with the instanton partition function of SU (2) × SU (2) (without matter fields) [6], finding complete agreement. We have also compared the result for SO (5) with N V = 1 with that of U Sp(4) with N A = 1 [6], again finding complete agreement. Going beyond the 2-instanton level is quite difficult. Taking into account the correction factor, the superconformal index takes the form, the subtraction can only be carried out at the one-instanton level, and has the form of (2.3), with the fundamental string state contributing charges in the fundamental representation of the U Sp(2M − 6) flavor symmetry, and the 3-pronged string contributing charges in the vector representation of the SO(M ) gauge symmetry. The analysis for higher instanton level is made complicated by this gauge charge. See the Appendix for details. For SO(4) with N V = 1, we compared with the 1-instanton partition function for SU (2) × SU (2) with a bi-fundamental [7], and find complete agreement. Including the corrected 1-instanton contribution, the superconformal index is given by . But this also requires the existence of flavorsinglet currents carrying two units of instanton charge, which we are currently unable to demonstrate. Both of these enhancements agree with the results of [26] using simplified instanton analysis.

Duality
As before, we can ask whether the continuation past infinite coupling yields a dual gauge theory. In the case of the O7 + plane we cannot describe the S-dual webs in terms of gauge theories, since we do not have a simple case in which there is a known dual gauge theory description. However we may still be able to provide an alternative formulation of the theory in terms of gauging a subgroup of the global symmetry in a lower rank SCFT, which in turn has an IR gauge theory description.
Let us take the SCFT corresponding to SO (6) with N V = 2 as our starting point. The global symmetry of the UV fixed point is SU (2) × U Sp(4) F , with the SU (2) realized nonperturbatively in the gauge theory. In the 5-brane web, Fig. 9a, the SU (2) is associated with the pair of parallel external NS5-branes. Gauging it is described by the 5-brane web of Fig. 9b. This corresponds to a SCFT with an IR description as SO (8) with N V = 2. So although we do not have a dual gauge theory for the latter, we can describe it alternatively as an SU (2) gauging of a rank 3 SCFT which has an IR description as SO (6) with N V = 2. This can be generalized in a straightforward way to SO(2N ) with N V = 2N − 6, leading to an alternative description as the linear quiver with gauge group factors SU (2) N −3 , where the last factor gauges the SU (2) part of the global symmetry of the aforementioned rank 3 SCFT.

SU (N ) with an antisymmetric
Incorporating matter in representations other than the fundamental or bi-fundamental in 5-brane webs is notoriously hard. Even in the simplest example of U Sp(2N ) with an antisymmetric hypermultiplet (corresponding to the rank N E 1 orẼ 1 theories), where the 5-brane web is known, the existence of the antisymmetric field is only understood indirectly, by analyzing the symmetry on the Higgs branch [27] (In fact there is a remaining puzzle in this case which we will review and resolve below). Using the same strategy, we also proposed a 5-brane web for SU (2N ) with an antisymmetric hypermultiplet in [18]. 9 Here we will give a more direct construction using the O7 − plane, which makes the antisymmetric hypermultiplet manifest. The construction involves a fractional NS5-brane that is stuck on the O7 − plane. Analogous constructions exist in 4d [15,28].

SU (2N )
The orientifold 5-brane web for SU (2N ) with N A = 1 is shown in Fig. 10a. Now different values of k correspond to distinct theories since the NS5-brane is not invariant under T ∈ SL(2, Z). The integer k is the CS level of the theory. Note that for a vanishing CS level, k = 0, the web is reflection symmetric, corresponding to charge-conjugation symmetry (which the CS term breaks). The hypermultiplet in the antisymmetric representation corresponds to an open string connecting the D5-branes on either side of the NS5-brane. The position of the external NS5-brane in the plane transverse to the O7 − plane corresponds to the mass of the hypermultiplet, and its position along the O7 − plane corresponds to its VEV, namely to the Higgs branch of the theory. On the Higgs branch the web reduces to that of U Sp(2N ), as it should. The orientifold web with the NS5-brane appears to exhibit 2N Coulomb moduli, since it has 2N faces. However they are not independent. There is one constraint coming from the fact that the NS5-brane cannot detach from the O7 − plane. The number of independent parameters is then 2N − 1, the dimension of the Coulomb branch of SU (2N ).
To our knowledge, the resolution of the O7 − plane with a fractional NS5-brane has not been previously studied, but one can make a conjecture based on the resolution of the bare O7 − plane. The only possibility that makes sense is that the combination is resolved into a (2, 1) 7-brane and a (0, −1) 7-brane with the NS5-brane ending on it (Fig. 10b). Using this, together with a couple of HW transitions and a T −1 transformation, we arrive at the web shown in Fig. 10c. This is precisely the web originally proposed in [18]. As an aside, we can now resolve two puzzles from [27]. The first has to do with the 5-brane web resolution of the classical brane configurations for the theories corresponding to the orbifolds without vector structure. For example for the Z 2 orbifold the theory is SU (2N ) 0 with N A = 2. The classical Type IIB brane configuration consists of 2N D5branes on a circle, with two O7 − planes at antipodal points, each supporting a fractional NS5-brane, Fig. 11a. We can now identify the resolved configuration, Fig. 11b, and using manipulations similar to the ones above, obtain a 5-brane web construction, Fig. 11c. The second puzzle has to do with the 5-brane web construction of the U Sp(2N ) theory with N A = 1 (Fig. 12). While this web exhibits the correct Coulomb and Higgs branches, it does not admit a deformation corresponding to giving a mass to the hypermultiplet. It exhibits only one mass parameter, corresponding to the YM coupling. We can now provide an alternative 5-brane web for the same theory, which does admit the hypermultiplet mass deformation. If we attach a fractional NS5-brane to one of the O7 − planes the theory remains the same; it is still U Sp(2N ) with N A = 1. This is the "Z 1 " orbifold. However the 5-brane web obtained from resolving this configuration is different, and now admits an extra deformation corresponding to the mass of the antisymmetric hypermultiplet (Fig. 13).

SU (2N + 1)
The orientifold web for SU (2N + 1) with N A = 1 is obtained by adding a fractional D5brane ending on the fractional NS5-brane, Fig. 14a. Note that this does not affect the NS5-brane, since the net number of D5-branes ending on it vanishes. This configuration would not be possible in the absence of the fractional NS5-brane; the consistency conditions of [29] would require an even number of D5-branes. The integer k again determines the CS level of the theory, which must now be half-odd-integer due to the parity anomaly associated with the antisymmetric matter multiplet. 10 It is important here that the fractional D5-brane is "broken" on the fractional NS5brane. In particular, the fractional NS5-brane can now detach from the O7 − plane by combining with the fractional D5-brane, as shown in Fig. 14b, giving an extra modulus for a total of 2N , in agreement with the dimension of the Coulomb branch for SU (2N + 1). This also implies that there is no Higgs branch, since the NS5-brane cannot be separated from the fractional D5-brane. This is also consistent with the gauge theory. For SU (2N ) the Higgs branch is spanned by the baryonic operator α 1 ···α 2N A α 1 α 2 · · · A α 2N −1 α 2N . For SU (2N + 1) there is no such gauge invariant operator. 11 There is also an obvious proposal for the resolution of the O7 − plane in this case, Fig. 14c.

Flavors
We can again add matter in the fundamental representation by attaching external D5branes. The condition of [3] for a fixed point to exist for SU (M ) with N A = 1 is N F + 2|κ CS | ≤ 8 − M . In this case we find a very different bound. The orientifold 5-brane web construction allows N F + 2|k| ≤ M + 5. The CS level is either k or k ± 1 2 , depending on whether N F + M is even or odd, respectively. In particular we can have an arbitrarily large rank. The bound is saturated by the web with the avoided intersection shown (for M = 2N and k = 0) in Fig. 15c. The two other interesting cases are N F + 2|k| = M + 3 and N F + 2|k| = M + 4, shown (again for M = 2N and k = 0) in Figs. 15a and 15b, respectively. These have parallel external NS5-branes. In all three cases we exhibit some of the extraneous instanton-charged states that should be removed from the instanton partition function. We expect these three theories to exhibit enhanced global symmetries in the UV, and in particular we expect the non-abelian flavor symmetry to be enhanced in the maximally flavored theory. In fact, these theories were recently studied in [30], using simplified instanton analysis, whose finding is indeed consistent with the enhancement suggested by these webs.

Duality
In formulating duality conjectures in this case we will follow the strategy we used for the U Sp(2N ) theories. We begin with a theory whose dual we know, and which we can engineer as a 5-brane web, and then gauge part of its global symmetry (and maybe add flavors) to obtain a new theory, whose dual we infer from the S-dual web.

Example 1
In our first example we begin with the self-dual SU (2)+6 theory. This is an IR gauge theory description of the rank one E 7 theory. The antisymmetric matter multiplet is neutral in this case. The web for this theory and its S-dual in the present description are shown in Fig. 16. The dual web makes manifest an SU (3) subgroup of the global symmetry, acting on the three flavors corresponding to the three external D5-branes. Now consider the 5brane web shown in Fig. 17a. This corresponds to gauging this SU (3) and adding one SU (3) flavor. The resulting gauge theory is the linear quiver 2 + SU (3) 0 × SU (2) + 3. The extra SU (3) flavor (other than the one we added) corresponds to the fractional D5-brane. The fact that the CS level is zero follows from the symmetry of the web. We can now read-off the dual gauge theory from the S-dual web, Fig. 17b, as SU (4) 0 with N A = 1 and N F = 6. We can generalize this construction by adding two more SU (3) Figure 16: The orientifold 5-brane web for SU (2) with N A = 1 and N F = 6 and its S-dual, which describes the same theory (we have taken k = 0).

Example 2
Our second example is slightly more intricate. We start with the self-dual SU (2)+5 theory (the IR gauge theory description of the rank one E 6 theory), described here by the web and its S-dual shown in Fig. 19. First we gauge the SU (2) ⊂ SO(10) F associated with the two flavors given by the external D5-branes on the left of the S-dual web, and add one flavor, Fig. 20a. This results in the quiver 2 + SU (2) × SU (2) + 3, where the extra flavor on the left comes from the fractional D5-brane as before. As the next step, we would like to gauge the SU (2) global symmetry associated with the two external (1, 1)5-branes. This gives the web shown in Fig. 20b, with the S-dual shown in Fig. 20c. The latter describes SU (4) ±1 with N A = 1 and N F = 6. But what theory is this theory the dual of? This is not immediately obvious since the SU (2) that we gauged is not a subgroup of the perturbative flavor symmetry of the quiver theory, which is [SO (6) In fact the quiver theory corresponds to a known 5d fixed point, the so-called R 0,4 theory, in which the factor SO(6) × SO(4) × U (1) 2 is enhanced to SU (8) [18]. 12 The SU (2) that we are gauging is therefore contained in the SU (8) factor. The R 0,4 theory has another IR gauge theory description as SU (3)1 2 +7 [18]. In this case the perturbative global symmetry is SU (7) × U (1) 2 , with one U (1) getting enhanced to SU (2), and SU (7) × U (1) getting enhanced to SU (8). We can then use SU (8) transformations to rotate the gauged SU (2) into the non-Abelian flavor symmetry of either of the gauge theories. This leads to three possible quiver theory duals of the SU (4) ±1 theory with N A = 1 and N F = 6: SU (2) × SU (2) × SU (2) + 3, 2 + SU (2) × SU (2) × SU (2) + 1, and SU (2) × SU (3)1 2 + 5. We can generalize this to higher rank as follows. Start with SU (2N )1 2 with N A = 1 and N F = 2N + 3. The 5-brane web describing this theory is a straightforward generalization of Fig. 19a. Performing the same manipulations on the web as before leads to a web that describes SU (2N + 2) with N A = 1 and N F = 2N + 4, generalizing Fig. 20c. Let us now find the dual theory. The original gauge theory corresponds to the UV fixed point theory R 1,2N +1 , a SCFT with global symmetry given by SU (2N + 3) × SU (2) × U (1) 2 [18]. Comparing with the gauge theory, we see that the topological U (1) symmetry is enhanced to SU (2) at the fixed point. The first step is to gauge the SU (2) and add one flavor, but we cannot do this in the gauge theory since the SU (2) cannot be rotated into the flavor symmetry. However we can use the dual gauge theory for R 1,2N +1 given by the quiver 2 + SU 12 The 5-brane web description in [18] is presumably related to the one here by resolving the orientifold plane. 13 For N = 1 this reduces to the previous example, since R 1,4 and R 0,4 are the same theory.  Figure 19: The orientifold 5-brane web for SU (2) with N A = 1 and N F = 5 and its S-dual, which describes the same theory.

Example 3
Next let us give two examples that involve SU (2N + 1). Start with SU (3) 0 with N A = 1 and N F = 7, Fig. 21a. This gauge theory corresponds to a 5d SCFT known as S 5 , which has an enhanced SU (10) global symmetry [18]. Gauging a specific SU (3) ⊂ SU (10) and adding an SU (3) flavor, we arrive at SU (5) 0 with N A = 1 and N F = 7, Fig. 21b. We can now use reasoning similar to before to derive a dual for this theory. The S 5 theory has two dual IR gauge theory descriptions. The first is SU (3) 0 with N F = 8, which is equivalent to our starting point since the rank two antisymmetric and fundamental representations are equivalent for SU (3). The other is the quiver 3+SU (2) Generalizing to higher rank, we find that SU (2N + 1) 0 with N A = 1 and N F = 2N + 3 corresponds to gauging an SU (3) in the S 2N +1 SCFT and adding an SU (3) flavor. The latter is a UV fixed point with several IR gauge theory manifestations, one of which is SU (2N − 1) 0 with N A = 1 and N F = 2N + 3 [18]. The full global symmetry at the fixed point (for N > 2) is SU (2N + 3) × SU (3) × U (1), and the SU (3) factor is precisely what is gauged. As before, we can obtain a dual of our starting theory by embedding SU (3) into

Example 4
Our second SU (2N +1) example is again more intricate, and will involve a two-step gauging of a SCFT. The starting point is the R 1,4 theory, which has a gauge theory realization as SU (3) 1 2 with N A = 1, N F = 6 [18]. The global symmetry of the gauge theory is SU (7)×U (1) B ×U (1) I , since the antisymmetric and fundamental representations of SU (3) are equivalent. The full global symmetry at the fixed point is SU (8) × SU (2). (Actually R 1,4 is identical to R 0,4 , but we will use the first description since the more general case below will involve R 1,N .) The 5-brane web for the gauge theory is shown in Fig. 22a. Now we gauge an SU (2) ⊂ SU (8), and add one flavor, resulting in the web shown in Fig. 22b. On the other hand, using SU (8) we can rotate the gauged SU (2) into the flavor SU (7) symmetry of the IR gauge theory, yielding 1 + SU (2) × SU (3)1 2 + 5 (which we can sort of understand from the S-dual of the web in Fig. 22b). We recognize this quiver theory as an IR gauge theory description of the SCFT R 1,5 [18].
Next, we would like to gauge the SU (2) corresponding to the parallel external (1, −1)5branes in Fig. 22b, leading to Fig. 22c. This corresponds to the gauge theory SU (5) 1 with N A = 1, N F = 7. As in example 2, this SU (2) is not contained in the flavor symmetry of the quiver gauge theory. In fact it is a subgroup of the SU (3) part of the full SU which is an IR gauge theory description of R 1,2N . The 5-brane web is a simple generalization of Fig. 22a. The full global symmetry of the SCFT (for N > 2) is SU (2N + 2) × SU (2) × U (1) 2 , whereas that of the gauge theory is SU (2N + 2) × U (1) 3 . In this case just the topological U (1) symmetry is enhanced to SU (2). The two-step gauging procedure leads to a web describing SU (2N + 1) 1 with N A = 1, N F = 2N + 3. The dual theory can be determined by considering the dual description of R 1,2N as the quiver 2 + SU (3)1 [18]. Gauguing the SU (2) flavor symmetry and adding a flavor leads to 1 + SU (2) × SU (3) N −1 + 5, which corresponds to the SCFT R 1,2N +1 . The other gauge theory description of this theory is in turn provided by 2 + SU (3) N −2 × [SU (3) + 1] × SU (2) + 3. Upon gauging the flavor SU (2) symmetry in this description we finally get A summary of all the general rank dualities discussed in this section is shown in figure  23. These dualities appear to be related to the 4d duals of SU (N ) with N A = 1 and N F = N + 2 [31]. It would be interesting to study this further using, for example, the superconformal index.

SU (N ) with a symmetric
To engineer a symmetric hypermultiplet we replace the O7 − plane with an O7 + plane. The resulting 5-brane webs for SU (2N ) k and SU (2N + 1) k± 1 2 with N S = 1 are shown in Fig. 24. In this case the maximal number of flavors that one can add for SU (M ) is N F = M − 3, which corresponds, as before, to an avoided intersection of external 5-branes. Also as before, for N F = M − 3, M − 4 and M − 5, there are extraneous instanton-charged states that have to be removed from the partition function.
Like in the case with an antisymmetric, the bound we find is very different from the one in [3] where these gauge theories were ruled out. However, in the webs we find one can take the limit where all the branes intersect the O7 + plane, corresponding to the fixed point. This strongly suggests that these theories do exist as 5d fixed point theories.
As in the case of the SO(M ) theories, the S-dual webs in the present case do not seem to correspond to gauge theories, but we can use them to relate the original gauge theory to a partial gauging of a lower rank SCFT which does have a gauge theory description.

Conclusions
In this paper we explored 5-brane web constructions in the presence of an orientifold 7plane. We have shown that these can be used to argue the existence of new fixed points, motivate symmetry enhancement and duality relations, and to assist in index calculations. We have concentrated on a simple class of examples, mainly webs with only one full NS5brane. Yet, this method can be used also to construct more complicated webs with an arbitrary number of NS5-branes. This then allows us to describe a large class of SU (N ) linear quiver theories with edge groups U Sp(2N ), SO(M ), or SU (N ) with a hypermultiplet in the symmetric or antisymmetric representation. The studies done here can be straightforwardly generalized to these cases as well.
Besides adding an O7-plane we can also add an O5-plane, parallel to the D5 branes, without breaking supersymmetry. This also allows a realization of U Sp and SO gauge theories. Furthermore, this method should also allows realization of SO and U Sp alternating quiver, which to our knowledge have not been studied in 5d. Thus, it will be interesting to examine these constructions as well [32].
The study of the brane realization of 5d gauge theories and comparing with field theory data have also lead us to conjecture new results in string theory. Particularly, we expect that there are two varieties of the O7 − plane connected by an SL(2, Z) T transformation. It will be interesting to see if this can also be understood from the string theory perspective.
Poles at zero (or infinity) lead to the violation of the x → 1 x symmetry (required by conformal invariance) in the partition function. In general, for a single integral (e.g. for one SU (N ) instanton), For two integrals (e.g. two SU (N ) instantons) we have These poles are associated with extraneous states, whose contributions must be removed from the instanton partition function. The restoration of x → 1 x symmetry serves as a useful test of this procedure.
In this case we deal with gauge groups SO(N ) and Sp(N ) and consider only noncomplex representations. As a result the residues at zero and infinity differ solely by a sign (this can be seen by the u → 1 u invariance of the expressions for these groups). Thus, we can simplify to: and completely ignore poles at infinity.

A.1 U Sp(2N )
The k-instanton partition function for U Sp(2N ) has two components, Z + and Z − , corresponding, respectively, to summing over holonomies of determinant +1 and −1 in the dual gauge group O(k). The latter can be viewed as the sector with one gauge QM instanton corresponding to the non-trivial element of π 0 (O(k)) = Z 2 . For k = 2n + 1 there are n independent holonmies, and therefore n contour integrals, in both cases. For k = 2n there are n independent holonmies of determinant +1, but only n − 1 of determinant −1. The total partition function is given by Z + ± Z − , where the relative sign depends on the value of the discrete θ parameter. The expressions for the contributions of the gauge multiplet and fundamental hypermultiplets to each component can be found in the appendix of [7]. For k = 1 there is no integral, and both components are invariant under x → 1 x .
For k = 2 the determinant +1 component Z + involves a single contour integral, and there are poles at zero or infinity when N F ≥ 2N + 4. For N F = 2N + 4 we find that .
Therefore the combination This agrees with the 2-instanton correction term that we subtracted in section 2, eq. (2.3). For N F = 2N + 5 we find , and therefore the combination This agrees with the 2-instanton correction term, eq. (2.3), for this case.
For k = 3 both components of the partition function involve a single contour integral, and again poles at zero appear for N F ≥ 2N + 4. Let's concentrate on the case with N F = 2N + 4, for which we gave a proposal for the full multi-instanton correction factor in eq. (2.4). For N F = 2N + 4 we find implying that the combination is x → 1 x invariant. For k = 4, Z + has two contour integrals and Z − has one. Both involve poles at zero for N F ≥ 2N + 4. We focus again on the case N F = 2N + 4. Evaluating the double contour integral for Z 4,+ requires a little more work. We can separate the non-zero poles into mixed ones (where u 1 ∝ u 2 ) and non-mixed ones. The non-mixed ones come in identical pairs of (u 1 = 0, u 2 = 0) and (u 1 = 0, u 2 = 0). It is not difficult to see that where Z 2,+ [u 2 ] is the integrand of Z 2,+ . Combining this with Res[u 2 = 0, u 1 ] and with Z 4,− , and using (A.4) we find In deriving this we used: where the residues are for the 2 instanton integrand, Z 2,+ + Z 2,− . To this we need to add the contributions of the four mixed poles, which give . (A.12) Combining everything we find One can then show that the combination x . In fact the expressions in (A.6), (A.9) and (A.14) reproduce the multi-instanton correction factor for N F = 2N + 4, eq. (2.4), to instanton number four.

A.2 SO(M )
The dual gauge group for k instantons of SO(M ) is U Sp(2k). The expressions differ slightly between even and odd M . We denote M = 2N + χ where χ = 0 or 1. The contributions of the gauge and fundamental (vector) matter multiplets can be lifted from the 4d results of [33,34]. The gauge multiplet contributes , and fundamental hypermultiplets contribute As in the U Sp(2N ) theory, we expect the corrections to the higher instanton contributions for N F = M − 4 to sum to a simple form involving a plethystic exponential. The analysis of the 2-instanton contribution is similar to that of the 4-instanton contribution to Z + in the U Sp(2N ) theory. There are two contour integrals, and the contributions to the non-invariance can be split into a mixed and non-mixed parts. The non-mixed parts give 1 −

A.3 SU (N )
The contributions from the gauge multiplet as well as from flavor in the fundamental were given in [7]. As we do not us them in this paper, we have not reproduce them. However, we shall write the contribution of matter in the symmetric and antisymmetric representations of SU (N ). For the symmetric we can lift from the 4d results of [34], finding: The contribution for the antisymmetric of SU (N ) was already given in [7]. Unfortunately, there was a mistake there which we take the opportunity to correct. The correct contribution is: Both of these provide additional poles in the integrand. The prescription for dealing with them follows from the results of [9]. Specifically, one defines p = 1 zx and d = z x , calculates the integral assuming x, p, d << 1, and only at the end return to the original variables.
Note that as generically with matter contributions, the symmetric and antisymmetric provide negative powers of u a . Specifically, they behaves as Z k S [u a ] ∼ u − N ±4 2 a , where the + sign is for the symmetric and the − for the antisymmetric. Since a fundamental goes like u − 1 2 a , we see that as far as the lowest u power and thus the appearance of zero poles is concerned, a symmetric contributes as N + 4 fundamentals and an antisymmetric as N − 4 fundamentals.
Finally a note regarding signs. When evaluating SU partition function with fundamentals it was noticed that a sign shift of (−1) k(κ+ N f 2 ) is also needed. When working with symmetrics or antisymmetrics using (A.24,A.25) we expect this to generalize to (−1) k(κ+ i ) where the sum is over all the matter field, R i is the representation of the i matter field under SU (N ) and C 3 [R] is the cubic Casimir of the representation R (normalized so that C 3 [F ] = 1). For antisymmetrics, we can compare with other methods, and we have checked that this is indeed consistent with results obtained by a different formalism.